demonstration of a neutral atom cnot gate and perspectives...

Demonstration of a neutral atom CNOTgate and perspectives for multi-qubit operations

Mark Saffman

Neutral atom quantum gates

Quantum gates via Rydberg blockade

Scalability

Preparation of single atom states

Collective encoding

two trapped atoms

People

AlsoAlex Gill (PhD) Jon Sedlacek, Jake Covey, Adam Beardsley (undergrads)

AlumniMarie Delaney Precision Photonics, BoulderPasad Kulatunga Hobart & William Smith Colleges, facultyDeniz Yavuz UW Madison, facultyTodd Johnson NIST, Boulder

CollaborationsKelvin Wagner UC Boulder – qubit addressing opticsJungsang Kim Duke – qubit addressing opticsKlaus Mølmer Aarhus - theory

Deniz Thad Erich MS Thomas Larry Yavuz Walker Urban Henage Isenhower

ToddJohnson -> Boulder

Neutral atoms in optical lattices

Say λ = 1 μm.

4 million lattice sites per mm2

8 billion lattice sites per mm3

Critical issues:• loading• addressability• interconnects

Neutral atom quantum gates

Many body collisional entanglement

Signatures of entanglement have been demonstrated in lattices, but not at the two-atom level:

C-phase gate = Uπ

21 32

1 3

Error scaling of Rydberg gate

residual excitation error ~Ω2/B2

Rydberg lifetime τ gives error ~1/Ωτ

optimum Rabi frequency Ω~B2/3/τ1/3

minimum error~ 1/(Bτ)2/3

Ω τ=1/γ

Rb atoms, n=100 and R=10 μm B~25 MHz, τ~300 μs

Error ~ 0.003

Saffman & Walker PRA (2005)

Experiment

MOT +

Experimental approach

+

0 atoms

1 atom

Optical traps (FORT)

Single atom detection

MOT +

Experimental approach

+

0 atoms

1 atom

Ground Rabi floppingΩ/2π=1.4 MHz

Ramsey measurement T2 ~ 1. ms

Optical traps (FORT)

Single atom detection

87Rb hyperfine qubit

|1,0>

|2,0>

6.8 GHz

Yavuz, et al. PRL (2006)

1F=25S1/2 |0>

|1>

F’=32105P3/2 ΔΩ1 Ω2

Microscopic optical trap

mx μσ 1~2

mKkUmKTmmw

Bfa 5/,5.01.006.1,0.3

=−=== μλμ

mz μσ 10~2

Single beam dipole trap -U ~ Intensity Pioneering experiments byGrangier,Meschede, groups

Rb λD2=.78 μm U

Atom temperature measured by drop and recapture. Typically T~200 μK

Tdrop/Tvibration

Pro

babi

lity

of re

tent

ion

2w~ 6 μm

Single atom readout with 3D molasses

Rydberg excitation geometry

|2,2>

|mj=1/2>

focused excitation beams, w=10 μm

3.4 GHz

= 11 G

Johnson et al., PRL 100, 113003 (2008)

Rydberg blockade between two atoms

Jaksch, et al. 2000

Rydberg blockade between two atoms

Z=10

μm

Jaksch, et al. 2000

Average of 146 shots.Probability of one atom in both sites about 5-10%.

Rabi oscillations

79d5/2

Rabi oscillations

79d5/2

Targeted site

Ω/2π= 0.51 MHz (exp.), 0.59 MHz (theory)

Rabi oscillations

79d5/2

Targeted site crosstalk at the other site

Ω/2π= 0.51 MHz (exp.), 0.59 MHz (theory)

Rydberg Blockade

site1 -> 2

no control atomtarget oscillations

with control atomtarget blockaded

Pg

79d5/2

Rydberg Blockade



site1 -> 2

Pg

T (μs) T (μs)0 1 2 3 4 0 1 2 3 4

79d5/2

Rydberg Blockade



site1 -> 2

Pg

site2 -> 1 P

g

T (μs) T (μs)

79d5/2

Higher fidelity blockade

We can do better by increasing the interaction strength

Interaction strength vs. n

B ~ n11 – n12

Predicted blockade and P2for n=90

Blockade experiment n=90

The data are fit to

With a control atom present we find

a= 0.09 (experiment) a=0.11 (simulation)

Residual blockade leakage is almost entirely due to state preparation and measurement errors.

Experiment Monte Carlo Simulation

Urban, et al. Nature Physics Feb. (2009)Gaetan, et al. Nature Physics Feb. (2009)

From blockade to quantum gates

The CNOT gate can be realizedwith a controlled phase plus Hadamards

From blockade to quantum gates

|1,0>

|2,0>

6.8 GHz

Better ground statecoherence with m=0 states at B~4 G.

Ground state Ramsey spectroscopy

The CNOT gate can be realizedwith a controlled phase plus Hadamards

~ 10 μs

Other CNOT protocols

Ohlsson, et al. Opt. Commun. (2002)

It is also possible to perform a CNOTgate using only π pulses and blockade:

Other CNOT protocols

Ohlsson, et al. Opt. Commun. (2002)

Experimental sequence

It is also possible to perform a CNOTgate using only π pulses and blockade:

00

97d5/2

F=2, m=0

F=1, m=0

CNOT experiment

ideal truth table

CNOT experiment

measured input states

Preliminary data May 2009

CNOT experiment

about 75 two-atom trials/ matrix element

measured truth table


CNOT experiment

54.)()4/1( exp =UUTr Tideal

fidelity

Probabilities

1/1703./54. =average high/low

about 75 two-atom trials/ matrix element

measured truth table


Entanglement on demand

210|01|1|

21|0| >+>

>⇒>+>

inputdata

ideal output

control target

Entanglement on demand

210|01|1|

21|0| >+>

>⇒>+>

inputdata

ideal output

control target

211|00|0|

21|0| >+>

>⇒>+>

inputdata

ideal output

control target

Publication in preparation Preliminary data May 2009

Scalability

Scalability

Ion traps N=8 qubits (2005)Neutral atoms Superconductors Semiconductor spinsPhotons and linear opticsNitrogen vacancies…..

Applications:

Entanglement and decoherence N> 2Quantum simulation N> 30Factoring, search, … N> 106

Quantum networks, secure data transmission

Neutral atom Rydberg gate array

2D array of optical traps

focused laser beamsfor state preparation,

single qubit operations,and measurements

hyperfine qubit

p

r


focused laser beamsfor Rydberg excitation and two qubit gates

Johnson, et al. PRL (2008)Urban, et al. NP (2009)

Neutral atom Rydberg gate array concept

focused laser beamsfor state preparation,

single qubit operations,and measurements

focused laser beamsfor Rydberg excitation and two qubit gates

hyperfine qubit

p

r


long range, notnearest neighbor

interactions

Johnson, et al. PRL (2008)Urban, et al. NP (2009)

Atomic interactions

Long range interactions between ground state atoms

10 μm

atom 1 atom 2

Rb-Rb magnetic dipole interaction

HzE μ25~Δ

Long range interactions between Rydberg atoms

10 μm

We have shown that this interaction can be controlled and used for manipulating quantum states of single atoms.

Rydberg n=100 van der Waals interaction

MHzE 25~Δ

2

2

4 dDN π

=

How many sites can be connected via Rydberg ?

How large can D be and how small can d be?

Interaction strength

How large can D be?

Numerical calculations for Rb, Cs in the range 50<n<100 show that

due to . Also we can tune δ with external fields.

So

At constant gate error E, and Rydberg lifetime τ

113

4422

666 ~

/1~~, n

nnndd

CDCUvdw δ

=

126 ~ nC

43 /~/1~ nkn δδδ →

5.0|~| ps νν −

( ) 3/76/16max ~~ nECD τ

Saffman and Walker, PRA (2005)

220 ~ nnττ =

Maximum number of connected sitesThe minimum separation is set by the requirement that a Rydberg atom does not collide with a ground state neighbor.So

Combining the above we find

Saffman and Mølmer, PRA (2008)

2201min ~ nnakd =

nnEmckk

N

nnEmckk

N

D

D

~43

~2

3

2/12/1

022

2/131

2/12/1)3(

max

3/23/23/13/1

022

3/121

3/8

3/2)2(

max

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

h

h

ταπ

ταπ

δ

δ

Connected qubits vs. gate error

Rb, n=100, dmin=1 μm

Challenges: optics, single atom loading,…

7600

470)3(

max

)2(max

=

=D

D

N

N

At E=0.001

A primary challenge is preparation of an array of singly occupied sites.

Single atom loading

A primary challenge is preparation of anarray of singly occupied sites.

BEC - MI transition + transfer to long period lattice (Phillips, Porto) adds complexity

Stochastic loading: does not use all sites

Collisional blockade: (Grangier) does not use all sites

Light assisted collisions in 3D lattice (D. Weiss) does not use all sites

The stochastic loading methods will enable small arrays provided one is willing to throw away half the sites.

Deterministic loading enabled by entanglement: (Saffman, Walker (2002))

Single atom loading

37.0~/11 eP ≤5.1 ≤P

5.1 ≤P

1~1P

Ensemble qubits

Qubits can be represented by single atoms or by ensembles using Rydbergblockade.

∑=j

NjN0..1..011 1N0....00 1=

“W” state with unit excitation symmetrically shared among all N atoms

Logical 0 Logical 1

Lukin, et al. PRL (2001)

Protocol for deterministic single atom loading.

Ensemble qubits and single atom loading

Protocol for deterministic single atom loading.

• load N atoms into optical trap• pump all N atoms to |0>• transfer “1” atom to |1> via Rydberg• eject (N-1) atoms in |0> using radiation pressure

∑=j

NjN0..1..01

1ψ


N0....01=ψ 1=ψ

Saffman and Walker PRA, (2002)

Preliminary results with this loading protocol have produced strongly sub-Poissonian Statistics.


<n>~3

before

Preliminary results with this loading protocol have produced strongly sub-Poissonian Statistics.


1

2

<n>~30

after

P0=.50 P1=.47 P2=.03

before

ran with <n>initial~6

Optimization in progress….

N-atom entanglement

The “W” states can be created very naturally with Rydberg blockade

∑=j

NjN0..1..01

1ψ

“W” states do not have the greatest possible degree of entanglement.Measuring one atom does not uniquely determine the state of all others.

A maximally entangled N atom state is the Schrödinger cat or GHZ state

This can also be efficiently prepared with Rydberg blockade.

( )NN 1....1110....0002

1+=ψ

Generation of N-atom Schrödinger Cat states

Basic idea - exploit asymmetric Rydberg interactions:

|0> |1>

|s> |p>strongΔss

strongΔsp

weakΔpp


Basic idea - exploit asymmetric Rydberg interactions:

|0> |1>

|s> |p>strongΔss

strongΔsp

weakΔpp

This is possible with a ratio > 150.

|s>=41s1/2|p> = 40p3/2R=3 μm


Prepare

N-atom cat !

strongweak

Generation of N atom Schrödinger Cat states

Prepare

Saffman & Mølmer PRL to appear, arXiv:0812.2425

Müller et al., PRL (2009) (N~3)

Numerical simulations and perturbation theoryshow fidelity of 85% for N=8.

strongweak

Fidelity of 8 atomcat state

Collective encoding of an N qubit register

K>=N atoms, each with N+1 states all in one site Less sensitive to atom loss – information is distributed.

Collective encoding of an N qubit register

E. Brion, K. Mølmer, and M. Saffman, Quantum computing with collective ensembles of multi-level systems,PRL (2007)

K>=N atoms, each with N+1 states all in one site Less sensitive to atom loss – information is distributed.

Assume K atoms each with N+1 long lived ground states |0>, |1>, |2>, |3>,.... |N>

Collectively encoded states

These states can be prepared starting from |0> using ensemble dipole blockade Lukin, et al. (2001).

Note fiducial register states are now entangled. For example

K0...00

K0...10 K

0...11

)002102012001001202102010

010201202100100210201200)(12/1(11004

++++++

+++++==K

Collective encoding in Ho

atom isotope J I ground laser cooling BEC

states demonstrated demonstrated

Yb 173 0 5/2 6 x xSr 87 0 9/2 10 x xRb 85 1/2 5/2 12 x xAl 27 1/2 5/2 12 xCs 133 1/2 7/2 16 x xTm 169 7/2 1/2 16Fe 57 4 1/2 18 xIn 115 1/2 9/2 20 xCr 53 3 3/2 28 x xSc 45 3/2 7/2 32Tb 169 15/2 1/2 32Ti 49 2 7/2 40

Mo 95 3 5/2 42Pr 141 9/2 5/2 60Tb 159 15/2 3/2 64Co 59 9/2 7/2 80Dy 161 8 5/2 102Er 167 6 7/2 104 xHo 165 15/2 7/2 128

With more internal states we can prepare a larger register.

Collective encoding in Ho

Ground state 4f116s2, J=15/2 has 128 hyperfine states

Encoding with two states per qubit gives a 60 qubit register.

E. Brion, K. Mølmer, and M. Saffman, PRL (2008)

Level structure in Ho

a-g cooling and trapping transitionss1, s2 shelvingr1,r2 readout

coolingreadout

Freezing out collisions

1.6 μm0.2 μm

Freezing out collisions

1.6 μm0.2 μm

Bottle beam, demonstrated with Cs

Isenhower, et al. Opt. Lett. 2009

1000 qubit scale processor

1000 qubit register based on:

18 sites, each with a 60 qubit ensemble with K~100 atoms/ensemble

Saffman and Mølmer, PRA (2008)

Summary

Rydberg blockade at >10 μm

CNOT gateentanglement

Single atom loading

collective encoding

fidelity=.54

demonstration of a neutral atom cnot gate and perspectives...

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